08/27/2007, 08:54 AM
So it indeed seems that Gottfried's method also leads to the same result as Andrew's approach. So I add to the list of lacking proofs
Let \( E_b \) be the power derivation matrix of \( x\to b^x \) at 0, show that \( E_b^t := \sum_{n=0}^\infty \left(t\\n\right) (E_b-I)^n \) exists (for each \( b>1 \) and each mxm truncation of \( E_b \)) and that the first row are the coefficients of a powerseries \( e_b^t \) such that \( \text{slog}_b(e_b^t(1))=t \) for Andrew's slog.
An easier step into that direction could be to show that \( e_b^t(x) \) is equal to the regular iteration at the lower fixed point of \( b^x \) for \( 1<b<\eta \).
Let \( E_b \) be the power derivation matrix of \( x\to b^x \) at 0, show that \( E_b^t := \sum_{n=0}^\infty \left(t\\n\right) (E_b-I)^n \) exists (for each \( b>1 \) and each mxm truncation of \( E_b \)) and that the first row are the coefficients of a powerseries \( e_b^t \) such that \( \text{slog}_b(e_b^t(1))=t \) for Andrew's slog.
An easier step into that direction could be to show that \( e_b^t(x) \) is equal to the regular iteration at the lower fixed point of \( b^x \) for \( 1<b<\eta \).